Title: “Modeling Coral Invertebrate Communities Using 3D photogrammetry”

author: “Journ Galvan” date: “4/23/2020” output: html_document —

Incorporate cleaned data taken in the field of coral associated fish and invertebrates

Data was filtered to only include invertebrates identified in coral colonies. These were identified down to the species level. Species richness, abundance and Shannon weiner diversity index was calculated.

Incorporate coral data

Cleaned data of coral colonies taken from the field and using photogrammetry techniques were combined into one dataframe. Many more and accurate morphological measurements could be acquired using photogrammetry than with traditional measurements taken *in situ. Common measurements of volume, surface area and available space known as interstitial space were calculated of each coral colonly.

Available space was traditionally measured using 2D measurements of five randomly selected branch distances and averaging them. In this experiment, we took methods used by Neil E. Doszpot et al. to calculate a 3D measurement of available space using convex hull geometry and the software estimated coral volume.

Because we did not take random branch distance measurements in the field, we used the software models to randomly select branch distances to be averaged.

Three other measurements were calculated from the coral models: convexity and sphericity which capture volume compactness and packing which captures how much of an objects surface area is situated internally versus externally.

Wide or tight branching corals were initially qualitatively determined. However, classifying corals with convexity values \(>= 0.5\) as “tight” and \(< 0.5\) as “wide” offers another method of classifying coral morphology. For the rest of these tests, we will classify “wide” and “tight” branching corals using this method.

Merge invertebrate data with coral morphological data.

Dataframes were combined and seperate columns created for log and square root transformations.

Histogram

Look for unimodal or bimodal curve in data for average branch distance, interstitial space and convexity. Compare these curves to branch distances defined in the field to branch distances defined by convexity measuremnt of 0.5

Visualize correlations

Non-transformed and transformed morphological measurements correlated using *pairs.panels().

Log transformed field and software estimated volume and log transformed software estimated SA and interstitial space are all highly correlated to eachother. Packing is also highly correlated to volume and SA.

Paired dependent t-test

Here, we want to compare the differences between our software and manual measurements of volume, height, length and width. Manual measurements for coral volume were estimated from an elipsoid.

Since the corals are not independent, that is, they were taken on the same coral but with different methods, we will consider a paired t-test with dependent samples. Height, length and width measurements are also subjective since they may have been taken from different areas on the coral. We will first check for normallity assumptions. Normality will be checked with a qqplot.

## 
##  Paired t-test
## 
## data:  dim_bind$sqrt_volume_pg and dim_bind$sqrt_volume_field
## t = -8.7044, df = 58, p-value = 4.091e-12
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -30.32779 -18.98700
## sample estimates:
## mean of the differences 
##               -24.65739
## 
##  Paired t-test
## 
## data:  dim_bind$height_pg and dim_bind$height_field
## t = -5.1446, df = 58, p-value = 3.32e-06
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -2.835609 -1.247079
## sample estimates:
## mean of the differences 
##               -2.041344
## 
##  Paired t-test
## 
## data:  dim_bind$length_pg and dim_bind$length_field
## t = -0.93279, df = 58, p-value = 0.3548
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -1.0463083  0.3811287
## sample estimates:
## mean of the differences 
##              -0.3325898
## 
##  Paired t-test
## 
## data:  dim_bind$width_pg and dim_bind$width_field
## t = -3.7184, df = 58, p-value = 0.0004538
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -1.973207 -0.592183
## sample estimates:
## mean of the differences 
##               -1.282695

We reject the null hypothesis that the true difference in means is equal to zero for all measurements except for length.

Comparing morphological measurements between tight and wide branching corals

Graphically analyze software estimated volume and elipsoid volume

The below graphs compare the software estimated volume, height, length and width measurements to the same coral’s manual measurements. A one to one line was fit to show how differences in measuremets vary with increasing coral size.

The below graphs compare the squareroot transformed software estimated volume to the same coral’s squareroot transformed elipsoid volume. A one to one line was fit to show how differences in measuremets vary with increasing coral size.

The below graphs represent how invertebrate abundance relates to both volume measurements. We will test how well our software estimated method of calculating volume and interstitial space relates to invertebrate counts. We will first assess \(p<0.05\) to test if the linear regression slope was significantly different from zero. Correlation of dependent and independent variables \(R^2\) was also included in the tables.

  Invertebrate abundance
Predictors Estimates std. Error CI p
Intercept 13.69 3.09 7.36 – 20.03 <0.001
elipsoid volume 0.00 0.00 0.00 – 0.00 0.043
Observations 29
R2 / R2 adjusted 0.143 / 0.111
  Invertebrate abundance
Predictors Estimates CI p
Intercept 0.31 -0.92 – 1.54 0.606
Log elispoid V 0.28 0.14 – 0.43 <0.001
Observations 29
R2 / R2 adjusted 0.372 / 0.348
  Invertebrate abundance
Predictors Estimates CI p
Intercept 11.55 5.54 – 17.55 0.001
software est. volume 0.00 0.00 – 0.00 0.004
Observations 29
R2 / R2 adjusted 0.275 / 0.248
  Invertebrate abundance
Predictors Estimates CI p
Intercept 0.44 -0.67 – 1.56 0.424
Log software est. V 0.30 0.15 – 0.44 <0.001
Observations 29
R2 / R2 adjusted 0.391 / 0.369

The below graphs represent how invertebrate species richness relates to both volume measurements. We will test how well our software estimated method of calculating volume and interstitial space relates to species richness. We will first assess \(p<0.05\) to test if the linear regression slope was significantly different from zero. Correlation of dependent and independent variables \(R^2\) was also included in the tables.

  Invertebrate richness
Predictors Estimates std. Error CI p
Intercept 6.25 0.86 4.49 – 8.01 <0.001
elipsoid volume 0.00 0.00 0.00 – 0.00 0.011
Observations 29
R2 / R2 adjusted 0.218 / 0.189
  Invertebrate richness
Predictors Estimates CI p
Intercept 0.19 -0.69 – 1.07 0.661
Log elispoid V 0.21 0.11 – 0.32 <0.001
Observations 29
R2 / R2 adjusted 0.391 / 0.369
  Invertebrate richness
Predictors Estimates CI p
Intercept 5.88 4.15 – 7.61 <0.001
software est. volume 0.00 0.00 – 0.00 0.003
Observations 29
R2 / R2 adjusted 0.291 / 0.265
  Invertebrate richness
Predictors Estimates CI p
Intercept 0.30 -0.50 – 1.10 0.448
Log software est. V 0.22 0.11 – 0.32 <0.001
Observations 29
R2 / R2 adjusted 0.405 / 0.383

When comparing elipsoid volume to software estimated volume for invertebrate richness, we see significant p-values for both as represented in the above tables. Software estimated volume measurements and invert. richness show a lower p value indicating a greater slope from zero as well as having a higher correlation \(R^2\) value. When data was log transformed, there was little to no difference in p-values or correlation. All slopes are less than zero suggesting a logistic growth model fit for invert abundance with increasing volume. When comparing the slopes for software volume \(b=5.051e-4\) and manual volume \(b=1.862e-4\), we see that software volume shows a more linear fit.

Graphically analyze software estimated space and avg. branch distance

Software estimated available space calculated from convex hull and coral volume was related to invertebrate abundance. The same was repeated for the traditional method of average branch distance.

Dependent and independent variables were also compared after being log transformed.

  Invertebrate abundance
Predictors Estimates CI p
Intercept 16.01 9.85 – 22.17 <0.001
IS 0.00 -0.00 – 0.00 0.250
Observations 29
R2 / R2 adjusted 0.049 / 0.013
  Invertebrate abundance
Predictors Estimates CI p
Intercept 0.56 -0.59 – 1.70 0.329
Log IS 0.28 0.13 – 0.43 0.001
Observations 29
R2 / R2 adjusted 0.354 / 0.330
  Invertebrate abundance
Predictors Estimates CI p
Intercept 17.06 5.45 – 28.66 0.006
Avg. branch width 0.60 -5.02 – 6.21 0.829
Observations 29
R2 / R2 adjusted 0.002 / -0.035
  Invertebrate abundance
Predictors Estimates CI p
Intercept 2.54 2.10 – 2.98 <0.001
Log avg. width 0.23 -0.42 – 0.87 0.476
Observations 29
R2 / R2 adjusted 0.019 / -0.017

Log transformed data appeared to not violate normality assumptions so was used in this analysis. Relating interstitial space to invertebrate abundance gave a signficant p-value while average width did not. The difference in p-values and \(R^2\) correlation values were significant. This may provide evidence to average branch width not adequately capturing available space for invertebrates when compared to a 3D software estimated measurement.

  Invertebrate richness
Predictors Estimates CI p
Intercept 6.83 5.11 – 8.54 <0.001
IS 0.00 -0.00 – 0.00 0.053
Observations 29
R2 / R2 adjusted 0.131 / 0.099
  Invertebrate richness
Predictors Estimates CI p
Intercept 0.33 -0.47 – 1.14 0.403
Log IS 0.21 0.11 – 0.32 <0.001
Observations 29
R2 / R2 adjusted 0.390 / 0.368
  Invertebrate richness
Predictors Estimates CI p
Intercept 6.50 3.17 – 9.83 <0.001
Avg. branch width 0.73 -0.88 – 2.34 0.361
Observations 29
R2 / R2 adjusted 0.031 / -0.005
  Invertebrate richness
Predictors Estimates CI p
Intercept 1.81 1.50 – 2.13 <0.001
Log avg. width 0.25 -0.22 – 0.71 0.285
Observations 29
R2 / R2 adjusted 0.042 / 0.007

Log transformed data appeared to not violate normality assumptions so was used in this analysis. Relating interstitial space to invertebrate richness gave a signficant p-value while average width did not. The difference in p-values and \(R^2\) correlation values were significant. This may provide evidence to average branch width not adequately capturing available space for invertebrates when compared to a 3D software estimated measurement.

Compare mean invertebrate abundance and species richness of tight and wide corals

Visualize data using boxplots.

Data was log transformed except for diversity so as not to violate normality assumptions. It appears that there is no difference in abundance, species richness or diversity.

Before conducting a two sample t-test we checked for normality using qq-plots and Shapiro Wilk test and a Levene’s test for equal variance.

With a \(p>0.05\) we fail to reject the null hypothesis that our data is normally distributed for wide branching corals. With a \(p<0.05\) we reject the null hypothesis that our data is normally distributed for tight branching corals. Because we are working with smaller data sets, we cannot use the central limit theorem.

With a \(p>0.05\) we fail to reject the null hypothesis that our data is normally distributed for tight and wide branching corals. Both groups of corals that have tight and wide branching morphologies have equal variance.

Because tight branching coral is not normally distributed, we could either perform a transformation or non-parametric test. We will try log transforming our data first. A column log transforming the number of inverts has already been created

With a \(p>0.05\) we fail to reject the null hypothesis that the difference between the means of the log transformed invert counts are different from zero.

Multivariable linear model

Fit a single variable Yi=β0+β1Xi+ϵi or multivariable linear model Y=β0+β1X1+β2X2+…+βnXn+ϵ to our data using different combinations of explanatory variables. Our response variable will be abundance of invertebrates. We will use AIC and R-squared values to determine the best fit model.

Log transformation appears normal. Log transformed volume, surface area, interstitial space, and packing were morphological measurements that had a positive correlation to number of cafi. Log transformed sphericity has a negative correlation to cafi. Because our data does not contain any 0 counts for invertebrate (meaning every coral contained an invertebrate population) we applied a log transformation to our count data.

Linear models will be fit to all corals as well as subset to tight and wide branching morphotypes.

VIF>10 will be removed from the dataset

We fail to reject the null hypothesis that our residuals are normally distributed.

A step AIC will determine a parsimonious model

Poisson GLM

We will compare our log transformed poisson general linear model with a quasi-poisson general linear model with non-log transformed counts.